Toss steak, 1/4 cup soy sauce, garlic, and 2 tablespoons lime juice in a large bowl. Marinate for 10 minutes.
Preheat grill or broiler to high heat; cook steak 2 minutes each side. Set aside.
Place broccoli in a steamer basket. Fill a large pan with water 2 inches deep; bring to a boil. Reduce heat to a simmer. Place steamer basket on top of pan. Cover and cook broccoli 5 to 6 minutes or until crisp tender. Let cool 10 minutes. Place the broccoli in a large bowl with mint, basil, and cilantro. Add cucumber slices.
Slice steak and add to salad. Combine chiles, sugar, remaining 1/4 cup soy sauce, and remaining 3 tablespoons lime juice in a bowl. Toss gently into salad. Pile salad onto serving plates and serve.